... will not work properly. If you are a musician, you know anything in this posting already. If you read it anyway, please feel free to correct me at points I might be wrong.
Why I started this thread: I find it interesting that harmony in music intervals can be expressed with very simple math. Understanding intervals gives a glimpse into the beauty of music. It cannot explain why a melody is nice, but it does reveal some parts of it.
The octave frequency ratio is 2:1. This is set in stone as we hear the same note regardless if we double or halve the frequency. C sounds like c or c', just in another octave, while for example E will sound as a different note.
The most simple ratio with a fraction is 3:2 = 1.5. This interval is also known as fifth. The fifth is fifth note from the base note, hence the name. The fifth for C is G. Try to play both notes together, it sounds good, now you know why.
The next-simplest interval after 3:2 is 4:3. The complementary interval of the fifth, which is needed to get to an octave, is 4:3, too, as 3:2 ("fifth") * 4:3 = 12:6 = 2:1. This interval is known as fourth, as it is the same like F, the fourth note of the base note C. We could use any other note as base but will stay with C for this posting.
We see a development for usable harmonic intervals: 2:1, 3:2, 4:3, ...
These are the ratios of the overtones to each previous overtone. Perfect overtones are generated by an idealized string. If it oscillates with 100 Hz (roughly G), it also generates vibrations at 200, 300, 400, 500, 600 Hz and so on. (Roughly notes g, d', g', b', d''.) These overtones are normally softer than the base tone, the loudness development of each overtone gives the whole tone its characteristics. So our ears and brains are trained to process overtone ratios. It is no wonder why we consider them harmonic.
Approximation of the first five overtones for G
If we follow up, we get the 5:4 ratio next, which is also known as major third. The next interval would be 6:5, which is dubbed as minor third (all ratios considered for just intonation.) Major and minor thirds stacked are giving us the fifth again, as it should be. (5:4 * 6:5 = 30:20 = 3:2.) This sounds good so far.
The major triad, which is a major and then a minor third, gives us the known triad. For C major it is C-E-G. The frequency ratio is like 4:5:6, a part of the overtone spectrum with 1:2:3:4:5:6:... ratio. As we hear the same note in any octave as the same note, :6 is the same like :3, and :4 the same like :2 and :1, so in a way we get the sound of 1:2:3:4:5:6. This is why a triad sounds good. Or why it should sound harmonic, I will come to the issues soon.
The distance of the fourth to the fifth is of course a second, or to be more specific, the major second. It is 3:2 / 4:3 = 9:8. The second is the second "full" note above the base note. 9:8 hence is the frequency ratio of a full note. Now the trouble begins. Two seconds are slightly greater than a major third, and six of them are greater than an octave. Three major thirds are smaller than an octave.
With few exceptions, stacked intervals will not lead exactly to the target interval. It just roughly fits. If we tune any instrument for a perfect triad for a given key, we will have to compromise other intervals and also other keys.
The interval between minor and major third should give us the minor second. If we use the aforementioned ratios, it comes out to 25:24 which is of course too small. In just intonation, the minor second is 16:15. But even then two minor seconds are not giving us a major second as we would expect.
This means, even if you have an instrument on which you can play any pitch, you will not be able to play with perfect harmony as stacked intervals will not completely fit. To make it worse, we have to consider the overtone spectra. While slight differences in the base tones are bad enough, the interfering overtones will create many more unharmonic ratios.
The notes are clear, but what about the actual frequencies?
Interestingly enough, there is another relationship between the 12 notes, entirely based on the fifth and the octave. If we begin with any note, lets say with C, and then get the fifth, and then the next fifth and so on, we will have 12 fifths in 7 octaves where the circle completes. This way we hit any note exactly once, though in different octaves. A full-size keyboard with 88 keys does allow to play the entire circle beginning with C contra octave or some subcontra notes.
In reality of course it is not a circle, as 12 perfect fifths are slightly greater than 7 octaves. Since we must obey the octave, as only the octave ensures that we hear the same note, we can tune the fifth slighly smaller than the perfect 3:2 ratio to get exactly 7 octaves with 12 slightly-tuned fifths.
This is done for the equal temperament: We devide an octave to 12 intervals of the same size. Each minor second is 2^(1/12) times pitched higher than the previous key. With this approach, we however have no simple, or even any rational ratios beside the octave. Any ratio is irrational and therefore cannot be harmonic. The fifth (and as the complementary interval also the fourth) will almost fit. But the major third for example is really messed up. However we are used to it as any modern grand and many other instruments use equal temperament.
In equal temperament, the tritone is an interval which is its own complementary interval. Other intonations have other definitions for the tritone but it always lies between fourth and fifth. Equal temperament renders the tritone the most useless interval possible. It does not sound harmonic, nor disharmonic, it just sounds like nothing.
It does not matter if we use the fifth 12 times to create all notes from the two most simple ratios, 2:1 and 3:2, or if we use somewhat more complex ratios like we find in overtone spectra, we will never be able to create a throughout harmonic interval definition which allows to stack intervals properly. The most baffeling thing for me is that the current – and probably best – solution is equal temperament which provides only irrational ratios beside the octave.
If you have an instrument which can be tuned, do you tune it to equal temperament or do you prefer to have at least some intervals in just intonation?
Yes, there are always compromises tuning instruments. Equal temperament is just the lesser of all evils probably, particularly since real music changes keys all the time. You could tune a piano to be more in tune for one key, but it's not like you can re-tune it in the middle of a piece.
My position is that, if equal temperament is really bothersome, hire some string and wind instrument players or vocalists. Those instruments are easily flexible enough in intonation that the correct ratios and tuning can be achieved on the fly with some listening and small adjustments. edit: though in practice, even world-class musicians get a little out of tune sometimes.
It's mostly just percussive (non-timpani) instruments that have this predicament.
edit2: Here is a musical problem.
Say you have a dominant major triad leading into tonic, e.g. G major chord in the key of C, resolving to C. To tune a major triad correctly, you need to tune the third (B in a G chord) down something like 14 cents compared to the equal temperament tuning. However, you're supposed to tune leading tones (B in the key of C) up higher so it sounds right.
I'm so used to equal temperament that other temperaments sound a little off to me. But that's because equal temperament is everywhere. Besides I play mostly with another instruments that can't be temperated so it would be a mess to play and to listen.
I can remember a couple years ago when a theory and composition major friend of mine was explaining this to me I wasn't sure if he was serious but we sat down and did the math and a lot of stuff has made more musical sense since to me.
Like the second post said this isn't nearly as big of an issue for wind or string instruments as you can adjust your pitch with embrasure or in my case playing the trombone by having "sharp" and "flat" note positions depending on which harmonic you're in relative to the key of C (for which the instrument is correctly tuned).
when I tune my guitar I usually tune it using tuner for A440 and fifths for E, D G, tuner for B and harmony for E. so... equal temperament I guess
Interestingly I do the same but tune B down from high E which I tune to the octave so my B string always feels a little out of tune cause I'm trying to force exact tuning even though the frets aren't set-up to do that.
Vocally I'm pretty sure most people (with a good sense of pitch) automatically hit the exact interval as the human voice is perfectly capable of "retuning" itself to a new key although I'll note that I know a few people with perfect pitch who get muddled by this stuff as they know exactly where say "g" is but because their singing in a particular key they have to consciously pull the note flat to adapt to the key. I think that would be super awkward.
On June 11 2011 00:38 Myrmidon wrote: edit2: Here is a musical problem.
Say you have a dominant major triad leading into tonic, e.g. G major chord in the key of C, resolving to C. To tune a major triad correctly, you need to tune the third (B in a G chord) down something like 14 cents compared to the equal temperament tuning. However, you're supposed to tune leading tones (B in the key of C) up higher so it sounds right.
What to do?
Live with the wrong intervals. Even if you could tune your instrument in real time, you cannot avoid wrong intervals unless you never play with sustain and too complex chords. The third is really off with equal temperament but as we have almost perfect fifths and fourth, which carry most of the harmonics at least for my ears, I can live with it.
We also know for example that Eb is actually not D#. We just play the same pitch for these two notes.
On June 11 2011 01:19 EdSlyB wrote: I'm so used to equal temperament that other temperaments sound a little off to me. But that's because equal temperament is everywhere. Besides I play mostly with another instruments that can't be temperated so it would be a mess to play and to listen.
Same for me.
Interestingly, a real grand sounds better than a digital grand (at least someone I can afford.) An acoustic grand does not allow you to play a pure note because the vibrating strings will have an effect on surrounding strings of other notes. If I play a triad on my Yamaha DGX 630 keyboard, the three notes overlie themselves for the chord. If I play a triad on a real grand, the notes merge into the chord, I just hear the chord and not three notes. It sounds great, even though it is less accurate than on the keyboard.
I've always had perfect pitch but have been way too lazy to identify intervals versus frequency, etc. People who know I have perfect pitch will ask what the frequency is, but I can only tell the individual notes or notes in a chord and cannot identify frequency (though supposedly one is supposed to be able to do that?). This is also part of the reason I sort of shy away from composition/theory, etc. It's sort of childish, but I just play to enjoy the music instead of knowing and appreciating the theory and the like behind it
On June 11 2011 00:38 Myrmidon wrote: Yes, there are always compromises tuning instruments. Equal temperament is just the lesser of all evils probably, particularly since real music changes keys all the time. You could tune a piano to be more in tune for one key, but it's not like you can re-tune it in the middle of a piece.
My position is that, if equal temperament is really bothersome, hire some string and wind instrument players or vocalists. Those instruments are easily flexible enough in intonation that the correct ratios and tuning can be achieved on the fly with some listening and small adjustments. edit: though in practice, even world-class musicians get a little out of tune sometimes.
It's mostly just percussive (non-timpani) instruments that have this predicament.
edit2: Here is a musical problem.
Say you have a dominant major triad leading into tonic, e.g. G major chord in the key of C, resolving to C. To tune a major triad correctly, you need to tune the third (B in a G chord) down something like 14 cents compared to the equal temperament tuning. However, you're supposed to tune leading tones (B in the key of C) up higher so it sounds right.
On June 11 2011 06:55 Kernkraft wrote: Play a good Instrument e.g Violin
You still cannot play with perfect harmony as some strings are still oscillating when you play a new note. You cannot have perfect intervals as they just don't fit.
On June 11 2011 00:38 Myrmidon wrote: edit2: Here is a musical problem.
Say you have a dominant major triad leading into tonic, e.g. G major chord in the key of C, resolving to C. To tune a major triad correctly, you need to tune the third (B in a G chord) down something like 14 cents compared to the equal temperament tuning. However, you're supposed to tune leading tones (B in the key of C) up higher so it sounds right.
What to do?
Live with the wrong intervals. Even if you could tune your instrument in real time, you cannot avoid wrong intervals unless you never play with sustain and too complex chords. The third is really off with equal temperament but as we have almost perfect fifths and fourth, which carry most of the harmonics at least for my ears, I can live with it.
We also know for example that Eb is actually not D#. We just play the same pitch for these two notes.
Sorry if I misworded this, but I think you misinterpreted my question. With many instruments it's practical to be adjusting the pitch in real time and that is what's done in practice (e.g. violin as pointed below above, or pretty much any string or wind instrument), which was covered earlier in my post.
Here I'm asking what the role of the leading tone is, for a dominant triad. Taken out of context, a leading tone should be tuned upwards (compared to equal temperament) to get the correct interval/tuning. Also taken out of context, a third in a major triad should be tuned down (compared to equal temperament) to get the correct interval/tuning.
If you have a dominant triad resolving to tonic, aren't the two objectives conflicting? This question isn't directly about tuning systems.
On June 11 2011 06:52 Z3kk wrote: I've always had perfect pitch but have been way too lazy to identify intervals versus frequency, etc. People who know I have perfect pitch will ask what the frequency is, but I can only tell the individual notes or notes in a chord and cannot identify frequency (though supposedly one is supposed to be able to do that?). This is also part of the reason I sort of shy away from composition/theory, etc. It's sort of childish, but I just play to enjoy the music instead of knowing and appreciating the theory and the like behind it
As I tried to explain in the OP, one cannot have perfect pitch for all intervals. Even if you restrict yourself to one key, lets say C major (which is a quite tough limitation) the major second is not always the same between any given note in the scale. You could tune the second from the base note (C) perfectly, but the distance between some other notes would be different.
Lets say you play the major scale with five major seconds and two minor seconds. When you are finished, you are slightly above the octave.
On June 11 2011 07:05 Myrmidon wrote: Here I'm asking what the role of the leading tone is, for a dominant triad. Taken out of context, a leading tone should be tuned upwards (compared to equal temperament) to get the correct interval/tuning. Also taken out of context, a third in a major triad should be tuned down (compared to equal temperament) to get the correct interval/tuning.
If you have a dominant triad resolving to tonic, aren't the two objectives conflicting? This question isn't directly about tuning systems.
I am still not 100% sure if I got you right.
I have to think this through:
If we want to play a major triad, we could tune it perfectly because in this case, the intervals do stack properly. (Major + minor third = perfect fifth.)
The interval for the leading tone is the major seventh (15:8). As you already wrote, the leading tone also is the fifth plus the major third (3:2 * 5:4 = 15:8.) This also fits if we use octave minus minor seconds (2:1 / 16:15 = 15:8).
But you want to play the dominant triad, meaning is the fifth, major seventh (which is the leading tone of the tonic), and (octave plus) major second, so we get 3:2, 15:8 and 18:8 (or 9:8 if we take out the octave.)
The intervals within the dominant triads are the same as for the major triad. If we use just intonation, we could play it right.
So does this mean our biological sense of pitch/tuning is inaccurate?
Actually it means that there is no rational number between 1 and 2, for which the n-th exponentiation (where n is a positive integer) could be expressed with 2^m (where m is another positive integer.)
In other words, you cannot stack x equal, but rational intervals to an octave. The crux is that we need both the octave as well as rational – and if possible, quite simple – ratios to have truly harmonic sound. But we only can approximate it.
The current common solution uses equal, but irrational intervals (save for the octave, of course.) This is problematic but has its advantages.
There exist a number of five-tone-systems. (For me, this is a mystery for itself. Pentatonic scales can be found in any culture. I did not have yet discovered why they are so common.) Our western music often relies on a system with 12 notes per octave. Other systems use 19 or even more notes per octave but all systems are just approximations for exact harmonic ratios.
Our western music which is based on 12 minor seconds, knows a number of scales. There are enharmonic scales possible (C# major = Db major) but depending on how you would construct the scales, they could actually be different. Only equal temperament allows perfect enharmonics. We trick our senses to be able to switch scales within a piece of music.
If you listen carefully while you play a grand you actually can hear that some pitches are slightly off. It is not a biological, but mathematical issue, though an interesting one as you will have to sacrifice some properties to gain others.
On June 11 2011 09:26 MozzarellaL wrote: I expected to see some mention of logarithmic scaling of frequencies in your discussion of harmonics.
I was disappointed.
He essentially did when referring to equal temperament tuning. i.e. exponential tuning.
F_relative = F_base*2^(n/12) where n is the number of semitones between F_relative and F_base.
Put this equation on a log base 2 scaled graph and it will look linear.
Also, it is implicit in his discussion of ratios. i.e. ratios are relative to a fundamental frequency, therefore as the fundamental frequency increases, the rate at which the fundamental some number of semitones relative to it changes at a logarithmic rate.
On June 11 2011 09:26 MozzarellaL wrote: I expected to see some mention of logarithmic scaling of frequencies in your discussion of harmonics.
I was disappointed.
I assumed this also. My band director has always told us these things but I never really understood the theory behind it until now, quite interesting. This is the reason only instruments where pitch can be altered whilst playing (aka Wind instruments) can play perfectly in tune together.
In Reality, %99.9 of western culture doesn't hear just intonation as 'in tune', nor any other system other than equal temperament. The reason is that we use, as stated above, an irrational system of minutely adjusting tones so that all are equal. This was a long process beginning with the harmonic series and just intonation by way of the Pythagorean system, and subsequently modified to accommodate the use of more than one key (and then many) as history progressed (Pythagoran, Meantone, Werckmeister, Kirnberger, etc..) until we reached perfect equal temperament.
As an interesting note, the study of these systems reveals a very interesting historical correlation to the music that these systems influenced. As an example, do we know why Beethoven's 3rd Symphony was nicknamed the Eroica? Well, it's because Eb major as a key was known as the heroic key, in fact, every key had a unique 'mood' because the tuning system created intervallic discrepancies between each key, giving each one a unique style and pathos. A large chunk of western history was influenced greatly by this, and the tuning systems that gave rise to those discrepancies.
Anyway, so I feel it's important that I should go and dispel some of the erroneous 'facts' in this thread, or clarify them further.
On June 11 2011 07:00 [F_]aths wrote: You still cannot play with perfect harmony as some strings are still oscillating when you play a new note. You cannot have perfect intervals as they just don't fit.
This is possible, but it requires a trained ear to be able to hear them. You're also not talking about pitch differences when talking about sympathetic vibrations. These timbral changes aren't going to affect the pitch nearly as much as you suggest (as in they won't) as the sympathetic vibrations only consist of overtones as part of that specific pitch's or pitch cluster's harmonic series. This only acts as a perceptual change, not an actual one. It's like adding reverb to a track that didn't have it before. The pitch hasn't actually changed at all, but the perception of that pitch may have based on the acoustical properties of the type of space used.
On June 11 2011 00:41 storm8ring3r wrote: when I tune my guitar I usually tune it using tuner for A440 and fifths for E, D G, tuner for B and harmony for E. so... equal temperament I guess
Depends, do you only tune at A440 or do you have a tuner that plays the pitch of all the other strings? If it's the former, then you actually tune in just intonation (or more accurately, you tune to perfect intervals), but the guitar's metal frets take care of the pitch adjustments into equal temperament.
On June 11 2011 06:43 [F_]aths wrote: We also know for example that Eb is actually not D#. We just play the same pitch for these two notes.
Only in 12-tet (equal temperament) is that the case, as you mentioned. Especially in just intonation because of the meandering pitch ratio's, enharmonics are critically important to distinguish. A really interesting thing a professor did for us in my undergrad was we took a small Palestrina, or Dufay work and plotted the pitch deviation from beginning to end (using JI). We started on A440 and depending on the piece the eventual A landed anywhere from 429 to 445. This assumes that you are horizontally making each pitch relationship a perfect one from one note to the next.
Tuning and Temperament is a fascinating subject, however I should state that simply grafting westernized music theory ideas onto another cultures music system is a really, really bad idea. Many times you will find that there is absolutely no correlation, or a very poor one, from one system to another especially when dealing with western culture vs. African, Indonesian or Asian cultures. In the African and Indian culture for example, there is no such thing as a time signature. Though each have very different functioning rhythmic systems they fundamentally have no concept that lines up with our system. Sure there are Tala's and cross-rhythmic structures where content converges, but those should not be considered time signatures in the way that western culture identifies them. Heterophony is a great example as it is integral to African musical culture, but until very recently (post 1900), it did not exist in western culture.
On June 11 2011 06:52 Z3kk wrote: I've always had perfect pitch but have been way too lazy to identify intervals versus frequency, etc. People who know I have perfect pitch will ask what the frequency is, but I can only tell the individual notes or notes in a chord and cannot identify frequency (though supposedly one is supposed to be able to do that?). This is also part of the reason I sort of shy away from composition/theory, etc. It's sort of childish, but I just play to enjoy the music instead of knowing and appreciating the theory and the like behind it
The average ability for one to discern a difference in pitch I think was ~3 Hz, which is less than 1/8th of a semitone in 12-tet. This really isn't distinguishable by the general populous, and even to many musicians. Only if you have perfect pitch, or have extensive experience with different tuning systems are these differences usually noticable.
On June 11 2011 07:06 [F_]aths wrote: As I tried to explain in the OP, one cannot have perfect pitch for all intervals. Even if you restrict yourself to one key, lets say C major (which is a quite tough limitation) the major second is not always the same between any given note in the scale. You could tune the second from the base note (C) perfectly, but the distance between some other notes would be different.
Let's say you play the major scale with five major seconds and two minor seconds. When you are finished, you are slightly above the octave.
Assuming you are restricting yourself to a fixed pitch system or instruments that are in 12-tet (there are many tuning systems and instruments that go beyond 12 tones/octave). With fixed pitch instruments (such as piano) this logic would hold true, but once you bring in a moving system this is no longer a hard and fast rule. That's not to say that the focal pitch in question isn't going to fluctuate a bit, but it's not like that's an impossibility and doesn't happen (or didn't).
So Just for emphasis, much of what's in the OP is right, but only conditionally, as if we are to assume that either the instrument or tuning system cannot move through linear space and is fixed in its position (and does't have microtonal capabilities). Here's a great example of a piece played in JI (it was definitely not supposed to be played in JI, but it's a good example of the tuning)
I'm not a musician and don't even know how to read music but I found this really informative; I study physics and it's nice to have a physical explanation that I can apply to music. I have a good ear and I've always wondered why certain instruments (when played live) sounded slightly "funny", I could never put my finger on it but I think this may explain it!
^^Bach Prelude and Fugue sounds really strange because of the sustains being too long...they are really disruptive to listening to the piece. Better example of just intonation is to listen to the Tallis Scholars. They are so awesome, and they sing Palestrina too.
On June 11 2011 15:44 Fontong wrote: ^^Bach Prelude and Fugue sounds really strange because of the sustains being too long...they are really disruptive to listening to the piece. Better example of just intonation is to listen to the Tallis Scholars. They are so awesome, and they sing Palestrina too.
It has nothing to do with the soundfont or the envelope being used, but the actual interaction of the pitches/overtones with one another that makes it sound so odd/jarring, although I would agree that the envelope is a little long, but that isn't why you perceive this as so disruptive. To be specific, what's making the timbres so odd to you are the interactions of the various beats between pitches, in that they are completely unfamiliar to your ear.
In 12-tet every semitone has the same ratio, 1.059. Here, beats should interact in an evenly out of tune manner. But in JI a semitone is closer to 1.066 so beats will differ depending on if the interval is a pure one or an inharmonic one (right, forgot to tell the OP that its not unharmonic, but inharmonic). To go a little further, you couldn't just add two semitones to get a M2 as this doesn't follow the harmonic series. In other words in JI, every interval class has its own unique ratio and therefore have a unique way of interacting with other pitch or pitches, this is why the beating or interaction of tones is so disruptive to the non trained ear, because these relationships aren't uniform whereas in 12-tet they would, but the point of JI isn't uniformity, its purity of intervals. This also reinforces what the OP was saying in that if you actually stack pure intervals on top of one another, your octave no longer becomes a pure interval, or 2:1 ratio, but is in fact a bit wider.
I can tell you that I hear this just fine, but that's because I have quite a bit of experience with JI and other tuning systems. Also the Tallis Scholars may predominantly sing early music, but they sing in equal temperament. The thirds are far to high, and their tuning is much too even to be JI. The reason it may sound more pure is that in choral singing many choirs actively do not use vibrato in order to strengthen pitch/harmonic relationships within a work. It also creates a more even uniformity in the overall sound and an makes it much easier to tune properly because you don't have to worry as much about an individuals voice interacting or fighting with another as much as if you were to add vibrato in there. That is another interesting discussion, but slightly off topic.
On June 11 2011 07:05 Myrmidon wrote: Here I'm asking what the role of the leading tone is, for a dominant triad. Taken out of context, a leading tone should be tuned upwards (compared to equal temperament) to get the correct interval/tuning. Also taken out of context, a third in a major triad should be tuned down (compared to equal temperament) to get the correct interval/tuning.
If you have a dominant triad resolving to tonic, aren't the two objectives conflicting? This question isn't directly about tuning systems.
I am still not 100% sure if I got you right.
I have to think this through:
If we want to play a major triad, we could tune it perfectly because in this case, the intervals do stack properly. (Major + minor third = perfect fifth.)
The interval for the leading tone is the major seventh (15:8). As you already wrote, the leading tone also is the fifth plus the major third (3:2 * 5:4 = 15:8.) This also fits if we use octave minus minor seconds (2:1 / 16:15 = 15:8).
But you want to play the dominant triad, meaning is the fifth, major seventh (which is the leading tone of the tonic), and (octave plus) major second, so we get 3:2, 15:8 and 18:8 (or 9:8 if we take out the octave.)
The intervals within the dominant triads are the same as for the major triad. If we use just intonation, we could play it right.
The distinction is between looking at music vertically (all the notes at any given time, like looking up and down a score) or horizontally (the progression of one line over time on a score).
I'm pretty sure that leading tones are generally not tuned 15:16, and in many cases looking at it in terms of a major 7th is incorrect, if that's not its role.
edit: I'm pretty sure this is a common example of contradictions in tuning, but it's been ages since I've taken any kind of theory and I never took anything remotely advanced anyway.
On June 11 2011 07:05 Myrmidon wrote: Here I'm asking what the role of the leading tone is, for a dominant triad. Taken out of context, a leading tone should be tuned upwards (compared to equal temperament) to get the correct interval/tuning. Also taken out of context, a third in a major triad should be tuned down (compared to equal temperament) to get the correct interval/tuning.
If you have a dominant triad resolving to tonic, aren't the two objectives conflicting? This question isn't directly about tuning systems.
I am still not 100% sure if I got you right.
I have to think this through:
If we want to play a major triad, we could tune it perfectly because in this case, the intervals do stack properly. (Major + minor third = perfect fifth.)
The interval for the leading tone is the major seventh (15:8). As you already wrote, the leading tone also is the fifth plus the major third (3:2 * 5:4 = 15:8.) This also fits if we use octave minus minor seconds (2:1 / 16:15 = 15:8).
But you want to play the dominant triad, meaning is the fifth, major seventh (which is the leading tone of the tonic), and (octave plus) major second, so we get 3:2, 15:8 and 18:8 (or 9:8 if we take out the octave.)
The intervals within the dominant triads are the same as for the major triad. If we use just intonation, we could play it right.
The distinction is between looking at music vertically (all the notes at any given time, like looking up and down a score) or horizontally (the progression of one line over time on a score).
I'm pretty sure that leading tones are generally not tuned 15:16, and in many cases looking at it in terms of a major 7th is incorrect, if that's not its role.
edit: I'm pretty sure this is a common example of contradictions in tuning, but it's been ages since I've taken any kind of theory and I never took anything remotely advanced anyway.
It's not a contradiction per se, but a matter of making compromises because of the theory involved. So let's look at your example again;
Example 1 (in a fixed pitch system and in strict JI):
The Dominant 7th chord will be built on the 5th scale degree which is the 3:2 ratio. From there the M3 = 15:8, then the m3 = 9:8, and finally the 7th = 4:3. So say we tuned to A JI and you want the E7 chord, then the notes are E[3:2] G#[15:8] B[9:8] D[4:3]. However this is a problem because these ratios do not represent perfect intervallic content. As we know, a true M3 is at 5:4 ratio, and 15:8 is != 5:4, so in this fixed pitch system, if you are using a chord that is not part of the root harmonic series (in this case A), your intervals will not be pure, ever (though in this case they are extremely close). The problem arises because you cannot tune on the fly to make perfect intervals, so this creates an odd color to your temperament, and irregular ratios.
Example 2 (with an instrument with a movable pitch system):
Same thing, you want E7 in AM. Well we have a fundamental difference now. In order to get a E7 in A all we have to do is take the 3:2 ratio, and build a dominant 7th chord on that note. What does that mean? Basically you will end up with a perfectly tuned dominant 7th, E[1:1] G#[5:4] B[6:5] D[16]. However, as you probably guessed, now we have another problem. The notes of the E7 will not resolve properly as now we have different (but perfect) ratios to deal with. The question then becomes, how do you resolve a problem that has no perfect solution?
Let's just focus on the m7th of the chord, in this case and Its easier to show this in Hz so let's do that. Say you start on A 220, the 5th, E, will be 330 and the 7th of EG#BD will be D 293.33. Now we resolve the D to C# like we are supposed to using 16:15 and we get 274.95. This is resolved perfectly, but we need to show how it relates back to the root. A perfectly tuned D in A JI is 293.26, if we move to the C# that relates to A we get 275. Notice how the C# is now just a little lower than it used to be?
E[330] G#[418.77] B[247.5] D[293.33] resolves to E[330] A[223.34] A[220] C#[274.95]
After looking at this you're probably thinking to yourself, but these are basically the same! Yea, they are, but notice how we now have two A's? Also note that the G# in JI is much higher than if it was fixed. Because of both of these instances, the G# pushes the A up about 3 Hz if we want to use perfect intervals (horizontally), but of course we can't have two different A's so one has to conform. The obvious choice is that in order to retain the best tonal structure the leading tone resolution has to be compromised in order to preserve the harmony, else we have two roots.
And that is the crux of every major temperament that had to deal with traditional western harmony, at some point you had to make a compromise in order to preserve the more important structure, whatever that might have been (horizontal or vertical).
equal temperament and just temperament are both inferior. Everything has been done with them, there is no room for doing anything new any more. The true scale of the elite is the Bohlen-Pierce 5-7-9 frequency ratio based scale.
E[330] G#[418.77] B[247.5] D[293.33] resolves to E[330] A[223.34] A[220] C#[274.95]
After looking at this you're probably thinking to yourself, but these are basically the same! Yea, they are, but notice how we now have two A's? Also note that the G# in JI is much higher than if it was fixed. Because of both of these instances, the G# pushes the A up about 3 Hz if we want to use perfect intervals (horizontally), but of course we can't have two different A's so one has to conform. The obvious choice is that in order to retain the best tonal structure the leading tone resolution has to be compromised in order to preserve the harmony, else we have two roots.
And that is the crux of every major temperament that had to deal with traditional western harmony, at some point you had to make a compromise in order to preserve the more important structure, whatever that might have been (horizontal or vertical).
Thank you, I think you made the issue clear: We cannot have a fixed tuned just intonated instrument. While we could play in just intonation with other instrument, we probably cannot play in an orchestra if other instruments play other melodies.
With computer technology you can adjust the tonality and introduce more just intonation even if the music modulates very quickly.
It will sound different but that's the point. Will it really be for the worse when actually in equal temperament every tone is off? And it will still not be 100% 'perfect' but it's the best we can do.
When I tune my violin viola the fifths are closer towards the A than they are in a tempered instrument.
I still remember my teacher telling me about the tunings and my then conductor talking about the circle of 5ths.
Another small caveat my accompanist told me is to tune the violin slightly higher than the piano. I've heard upper instruments trying to tune slightly lower or equal to the lower strings and I can safely say imho being a flat upper part is hell on the ears.
The problem with things other than equal temperment is that music is everywhere using that system. Using any other system (like Baroque systems) will commonly just sound horribly out-of-tune to the basic listener. We are used to equal temperment.
Further than that, with the worldwide popularity of pop music, equal temperment is becoming the standard across the globe. Other temperments will probably only be used when aiming for more archaic voices in music (like when people are mimicking the Gamelan or something).
As an interesting note, the study of these systems reveals a very interesting historical correlation to the music that these systems influenced. As an example, do we know why Beethoven's 3rd Symphony was nicknamed the Eroica? Well, it's because Eb major as a key was known as the heroic key, in fact, every key had a unique 'mood' because the tuning system created intervallic discrepancies between each key, giving each one a unique style and pathos. A large chunk of western history was influenced greatly by this, and the tuning systems that gave rise to those discrepancies.
Those moods for keys were not universal. Different composers used different keys depending on what they associated with the key. The tuning system led to discrepancies, but moods were certainly not unique.
I created synthesized recordings (using some C code) of a Bach chorale in three different tunings:
--equitempered --"exact" just temperament, in which every note has its frequency adjusted based on its harmonic function so that these whole-number ratios are always present --"wrong" just temperament, in which an instrument tuned to one key is used to play in another key (which sounds like shit)
On June 11 2011 12:52 wo1fwood wrote:As an interesting note, the study of these systems reveals a very interesting historical correlation to the music that these systems influenced. As an example, do we know why Beethoven's 3rd Symphony was nicknamed the Eroica? Well, it's because Eb major as a key was known as the heroic key, in fact, every key had a unique 'mood' because the tuning system created intervallic discrepancies between each key, giving each one a unique style and pathos. A large chunk of western history was influenced greatly by this, and the tuning systems that gave rise to those discrepancies.
I have to say, this is the biggest epiphany for me. I've been doing both mathematics and music for so long, but taken for granted (what I had been told) about all nth intervals being equal on a keyboard. I knew from just playing around on a keyboard that I liked certain chords more than others (differing only in their tonic), but I never had any idea why.
I am curious, however. How does this all apply to, say, brass instruments, who are necessarily built off harmonic intervals? I play the trumpet, and just for example, if I play everything with open fingers, these notes are theoretically all following an overtone series of the fundamental frequency of the horn. (right?). Does this mean then, that as my range gets higher and higher, I would end up being out of tune compared to a piano? And then that my (for example) C to E interval changes depending which octave I am at?
The crazy sorts of things you learn at Team Liquid.... gracias OP and all contributing smart peoples.
the physics in the sound is always good to know, but most professional musicians know that what is correctly in tune frequency-wise is not what is always correctly in tune to the human ear (some common examples include pushing the upper note in an octave slightly down to bring it "in tune," pushing the third slightly down in a major chord or slightly up in a minor chord, etc).
...which is the tempered tuning you guys are talking about, i guess. just small tricks of the trade that you learn from playing in serious orchestras. many touring solo pianists have the upper and lower ranges of their instruments tuned off from the machine -- i.e. the center of the keyboard is almost always in tune with whatever the machine is set to, then the upper range might be tuned lower than the machine and the lower range pushed up closer to the center etc.
the point is that what the machine says is correct is not always what will sound correct.
Both systems of tuning, and all other additional systems of tuning simply offer more possibilities for expression. Chopin and his generation would be amazed that our society plays almost entirely in Equal temperament. It really just comes down to what has been accepted as cultural norm, neither is right or wrong. But I think the worst approach is assuming musical validity in something based in physics, if in physics the scales seems irrational it does not make it musically unacceptable. Instead musicians should choose the system that offers them the correct expressive devices ex. 19th century tuning causes dissonance to be that must more effective and gripping.
I'm glad that knowledge of this is getting out there. Our culture seems to have simply locked down into an equally tempered mindset when that is not always the only option that is available.
I am a harpist and I prefer to avoid equal temperament for expressive reasons, however, it is sometimes necessity to use it, ie. orchestra.
All of the just-tempered scales stem from ratios of small prime numbers. The main question is how many prime numbers to include.
If you only use 2, then you get a series of octaves (2/1, 4/1, 8/1, etc.), which is a boring scale (no specific reason why it should be boring, but it's boring nonetheless).
If you use 2 and 3, this enables you to fill in the 'space' between the octaves, creating a more interesting scale with ratios like 3/2, 4/3, 9/8, 16/9, 27/16, 32/27, etc. This produces the Pythagorean scale, with its notable comma (a frequency ratio of 3^12 / 2^19 = 531441 / 524288 = 1.013643) after scaling by a fifth (3/2) twelve times. The Pythagorean scale of course works much better in some keys than in others.
The just-tempered scale most people refer to uses the prime numbers 2, 3, and 5. This enables the space between the octaves to be filled in with eleven very convenient ratios: minor 2nd = 16/15 major 2nd = 9/8 minor 3rd = 6/5 major 3rd = 5/4 perfect 4th = 4/3 diminished 5th = 45/32 perfect 5th = 3/2 minor 6th = 5/3 major 6th = 8/5 minor 7th = 9/5 major 7th = 15/8 This kind of scale sounds better in more keys than the Pythagorean scale.
Some versions even include the prime number 7, and use a frequency ratio of 7/4 for the minor 7th interval.
To much bla and no music ! .... first couple of sentences i was like ermmm .. just overtones ? cba to read the rest ... i always hated it when music gets mixed with TO much theory, and i still do till this day =( ... ofcourse there needs to be knowledge in theory but to much .. is to much ... comprende!?
On June 12 2011 23:47 DoubleReed wrote: Those moods for keys were not universal. Different composers used different keys depending on what they associated with the key. The tuning system led to discrepancies, but moods were certainly not unique.
Yeah, I probably should have been clearer on that. There was a general consensus so to speak, whether you were referencing Galeazzi, or Christian Schubert's, or Helmholtz' writings on the subject on what each key inherently was supposed to convey, though it is true as you stated, sometimes that could be highly subjective considering the composer writing the piece, whether it was following that consensus or more of a personal affectation supplanting what the former was (I'm going to try to ask an musicology prof. about this how much of a consensus and get back to you).
On June 11 2011 01:19 EdSlyB wrote: I'm so used to equal temperament that other temperaments sound a little off to me. But that's because equal temperament is everywhere. Besides I play mostly with another instruments that can't be temperated so it would be a mess to play and to listen.
Same for me.
Interestingly, a real grand sounds better than a digital grand (at least someone I can afford.) An acoustic grand does not allow you to play a pure note because the vibrating strings will have an effect on surrounding strings of other notes. If I play a triad on my Yamaha DGX 630 keyboard, the three notes overlie themselves for the chord. If I play a triad on a real grand, the notes merge into the chord, I just hear the chord and not three notes. It sounds great, even though it is less accurate than on the keyboard.
I think that's common knowlegde that acoustic pianos are tuned with many strings/keys not exactly in tune. That actually makes the chords sound richer. I have a Korg LP-350 at home and when I turn on the Werckmeister III scale the chords sound a little 'harsh'. I can't play more than one minute with that temperament. ^^
I think that this subject can be summarized like this: music is art. Music is life. Music is Nature. As such you can't simply treat it like a math operation. Because then music will be an automaton, a robot and it will sound 'artificial'. You need to have some 'imperfections'. Only then you'll hear Nature's true sound.
I'm also a musician (guess the instrument) and we have to continually adjust intonation on the fly.
However, when working on technique, we tend to stick to equal temperament for training consistency. This way you should already have a good feeling of how you'll need to adjust, without listening and reacting every single note.
I hadn't put a whole lot of thought into this, it's pretty crazy.
On June 13 2011 06:49 zinic wrote: To much bla and no music ! .... first couple of sentences i was like ermmm .. just overtones ? cba to read the rest ... i always hated it when music gets mixed with TO much theory, and i still do till this day =( ... ofcourse there needs to be knowledge in theory but to much .. is to much ... comprende!?
This is not too much theory, in fact, I focused on some rough principles only. Why is the fifth as important as it is? Because it is the simplest frequency ratio with a fraction.
On June 11 2011 12:52 wo1fwood wrote:As an interesting note, the study of these systems reveals a very interesting historical correlation to the music that these systems influenced. As an example, do we know why Beethoven's 3rd Symphony was nicknamed the Eroica? Well, it's because Eb major as a key was known as the heroic key, in fact, every key had a unique 'mood' because the tuning system created intervallic discrepancies between each key, giving each one a unique style and pathos. A large chunk of western history was influenced greatly by this, and the tuning systems that gave rise to those discrepancies.
I have to say, this is the biggest epiphany for me. I've been doing both mathematics and music for so long, but taken for granted (what I had been told) about all nth intervals being equal on a keyboard. I knew from just playing around on a keyboard that I liked certain chords more than others (differing only in their tonic), but I never had any idea why.
I am curious, however. How does this all apply to, say, brass instruments, who are necessarily built off harmonic intervals? I play the trumpet, and just for example, if I play everything with open fingers, these notes are theoretically all following an overtone series of the fundamental frequency of the horn. (right?). Does this mean then, that as my range gets higher and higher, I would end up being out of tune compared to a piano? And then that my (for example) C to E interval changes depending which octave I am at?
The crazy sorts of things you learn at Team Liquid.... gracias OP and all contributing smart peoples.
On a keyboard, Eb is no different from any other key. The different moods associated with keys either comes from specific composers or from the temperment causing different keys to have different intervals. On a modern keyboard all keys have the same intervals, so if you like certain chords more than others it does not have to do with temperment in any way. However, many composers still associate moods with keys, so its nothing out of the ordinary.
Also, if I'm not mistaken, pianos are tuned specifically out-of-tune as it gets higher and higher to make the chords sound less dull.
On wind instruments, keep in mind that lots of tuning is done with voicing of your airstream, but yes, the higher you go the tunings get more out of tune. But the higher you go, the more overtones there are. This is why trumpet players need to use triggers up higher as well as valves. If you watch pro Trombone players, for instance, they will have a slightly different "3rd position" depending on the note they're playing.
Yes, on a keyboard B# is C and C## is D and we also have no key characteristics left. An acoustic instrument could have some key characteristics even if tuned in equal temperament as the resonance of the body could influence the overtone spectra. But it would not be the commonly assumed characteristic.
If I am not mistaken, grands are tuned "wrongly" because of the stiffness of strings. The actual slightly wrong tuning leads to perceived right temperament (equal temperament, that is.)
EPT![[image loading]](http://i.imgur.com/jl7jh.png)
![[image loading]](http://i.imgur.com/vRItL.png)
]. However, as you probably guessed, now we have another problem. The notes of the E7 will not resolve properly as now we have different (but perfect) ratios to deal with. The question then becomes, how do you resolve a problem that has no perfect solution?
